The empirical rule is also known as the 3 sigma rule.

In addition, this rule helps you analyse the market and will tell you where the probabilities are high to enter and exit the market for maximum profits.

What is an empirical rule? …

In standard deviation, we already understood that it is the % deviation +/- from the average of the data set that we took.

When it comes to this rule, it talks about it further, and it tells what if there is a 2 times SD and a 3 times SD.

You can see a pendulam in the above image.

In addition, pendulam always moves away from the midle to the left or right and always comes back to the middle again.

Moreover, this process continuously repeats.

The rule that I am talking about in this article also works in a similar way.

The empirical rule, also sometimes called the 3 sigma rule or 68-95-99.7 rule.

In addition, it is a statistical rule.

Which states that in a normal distribution, almost all observed data will fall within three standard deviations of the mean or average.

In addition, the empirical rule predicts that in normal distributions, 68% of the observations will fall within the first standard deviation (µ ± σ).

Moreover, 95% of the times within the first two standard deviations.(µ ± 2σ).

Again, 99.7% within the first three standard deviations (µ ± 3σ) of the mean.

Where µ is called mean. and σ is called sigma.

What does normal distribution look like?…

See the above image; it is showing MBA grades from 60 to 96.

In addition, if we just plot the MBA grades on the Y axis, the chart will be like the one shown in the above image.

In addition, we can calculate the average grades and standard deviation for the above-given data.

Moreover, we need the mean or average and SD for plotting the bell curve, i.e., the normal distribution curve.

Now, you can see in the above image that the average is 78 and the standard deviation is 10.954451.

In addition, I have calculated the normal distribution values with the help of Microsoft Excel.

Now, we have the normal distribution curve that we are looking for.

In addition, the height of the curve started from 60 and reached 78, then started decreasing from 78 to 96.

Moreover, you can see that the area of the curve is much larger from 68 to 88.

Always remember that the curve area is equal to 100.

Now, let’s apply the first standard deviation to the above curve…

Standard deviation value = 10.954451.

And we know that the grade will move plus or minus one standard deviation from the mean 68% of the time.

So, if we add average and SD = 78 + 10.954451 = 88.9544512,.

and if we subtract SD from the average  =  78 – 10.954451 = 67.0455488.

Finally, the grade value will fall between values 67.0455488 and 88.9544512.

The orange colour shows the 68% area of the normal distibution curve.

In addition, from average 78, 34% area is on left-hand side of curve.

and 34% of the area of the orange area is on the right-hand side of the curve.

Finally, we say as per this rule that if a new student joins the MBA, there is 68% probability that he will get a grade in between 67 and 88.

And 34% chance of getting a grade in between 67 and 78.

Again, 34% chances of getting a grade in between 78 and 88.

We can plot areas for 2 sigma deviations and also for 3 sigma standard deviations.

But the probabilities are low.

We know that students getting a grade of 63 and a grade of 90 are both quite low.

So, the probabilities are also low, and the area will also be low in the curve.

Now, let’s calculate the 2 standard deviation probability.

If you add average + 2*S.D = 78 + 2*10.954451 = 99.08.

Similarly, subtract 2 standard deviations from the average = 78 – 2*10.954451. = 56.09

So, the grade value will fall between 56.09 and 99.08  95% of the time.

In other words, we can say that grade will fall between 56 and 68 (17.5% time).

In addition, always remember that on the left-hand side first comes 1 sigma area, i.e., 34%; next comes 2 sigma value, i.e., 17.5%; and 3rd comes 3 sigma value, i.e., 1.65% times.

Let’s understand it with numbers.

1 sigma = 68% of times, so 68% /2 = 34%.

So, 34% on the left-hand side of the area from the average.

and 34% on the right-hand side of the area from the average.

Similarly, you can calculate for 2 sigma also.

2 sigma SD = 95% of times. So, 95%/2 = 47.5% from the average

So, 47.5% probability left hand side and 47.5% of times on right hand side from the average.

Again, 3 sigma = 99.7% times, so 99.7%/2 = 49.85%.

So, 49.85% times on the left-hand side from the average.

And 49.85% of times on the right-hand side from the average.

Finally, all the probabilities above are from the average, i.e., from the mean.

But we can get the area % probability from any point to any point.

Like in this case study, we can get probability from values 61 to 65 also.

Unlike the average 78.

Now, let’s apply the 3 siga rule to our case study.

3 sigma will come if you add average to 3*SD = 78 + 3*10.954451 = 110.86

and 3sigma if you substract average with 3*SD  = 78 – 3*10.954451 = 45.13.

Imp Note…

You may be wondering why 45 grades and 110 grades are coming.

In addition, 45 may be there in MBA, but grade 110 will not exist, right?

This formula of statistics doesn’t know that there will be a grade above 100.

And this 3 sigma is a very unlikely probability.

You can see in the probability chart too.

and you know already it is only 1.65% of times possible. (I am not saying you will get a grade above 100.).

Now, we cannot plot the chart for it.

Because 1.65% falls between 45 and 56 on the left-hand side. and there are no values in the data set that we have taken for from 45 to 56.

How do we apply this empirical rule to our investment decisions? …

Let’s take a case study of the Hdfc small-cap fund and apply this 3 sigma rule to it.

We know that more profit or more return on investment comes if you invest at low and if you sell at high only, right?.

Here, in the above image, I have Hdfc small cap fund yearly returns from January 2020 to January 2025.

and the average return for this fund is 31.1%.

In addition, the standard deviation is 21.38%.

Let’s apply 1 sigma standard deviation as per this rule.

1 sigma rule = average + SD = 31.1% + 21.38% = 52.48% (on the right-hand side of the average).

and 1 sigma ruel average – 2*SD = 9.7%.

So, it says, 34% of times the return will remain between 31% and 52%.

and it is very clear it is not good to invest if the fund already gave 50% return in a year.

So, its time to sell if the fund already gave 50% ROI.

Again, it is saying if they gave only 31%, then there is a probability of getting an other 20%, which is 34%.

As it is near the average of 31%.

But do remember, the roi at 31%, i.e., at average, doesn’t guarantee that the roi will increase.

And it is telling you only a probability %.

Similarly, on the left-hand side, the roi will be in between 9% and 31%; the roi is 34% probability under the 1 sigma rule.

Financially, it will be a huge opportunity for you if you can get a 9% ROI opportunity under the 1 SIMGA rule for this fund.

Because from 9% to 52%, there is a 68% probability. That is a huge probability.

Let’s apply the 2 sigma rule, i.e., the empirical rule, to this fund…

2 sigma if you add average to 2*sd = 73.86%

2 sigma if you subtract 2*sd from average = 31.1% – 2*21.38% = -11%.

As per the 2 sigma rule, the roi will fall between -11% and 73.86% roi for 95% probability.

So, it will be a jackpot for you if you see the last one-year ROI of this fund at -11%.

And you can blindly invest at that point; there is not even a 5% chance for this fund going down from this -11% ROI.

Moreover, there is a probability of giving 63% roi, i.e., -11% + 73% = 62% roi, and the probability for it is 95%.

Finally, it is very clear that we all should invest when the roi is somewhere close to average roi, i.e., 31% in this HDFC small fund, and the probability of a positive roi from this level is 49.85%. and to give an other 20% roi is 34% probability.

and invest aggressively when the last one-year roi is somewhere -10%, as the probability of positive roi is 95% from this level.

But do not forget to sell this fund when the fund gives the last one roi of 52%.

Because roi always will come back to its mean, like I said at the beginning of the article about pendulum.

The best time to invest in the HDFC Smal Cap fund using this empirical rule. ?…

Best time to buy…

If you find the ROI for the last one falls on the left-hand side of average, then it will be good to invest in this fund.

Best time to sell…

Similarly, if you find the ROI for the last one falls on the right-hand side of the average ROI, then its good for you to sell this fund and exit.

 

Read about buying a home with a loan—is it good?

Also read the article Standard Deviation: All You Need to Know.

And read about HIGH ROI when required to recover the loss in the capital?

Also read an article about How to Beat One Crore with One Lach of Investment.

and read an article about Tithe—How it will make you poor?